Information Processing Method

ABSTRACT

An information processing method transfers information from a start face to an end face with a minimum local distortion by maintaining one-to-one correspondence between the original information on the start face and the transferred information on the end face. The method includes an operation of mapping information taken from a three-dimensional surface onto a rectangular plane, or vice versa, by dividing the start face into a plurality of divisional start faces and preparing divisional end faces that just fill the end face, then deforming each divisional start face to just fit a corresponding one of the divisional end faces, so as to maintain lines and points defining each divisional end face as lines and points also on the end face and to ensure that a first area ratio between each divisional start face relative to the entire start face and a second area ratio between each divisional end face relative to the entire end face is substantially equal.

CROSS REFERENCE TO RELATED APPLICATIONS

This is a continuation-in-part of International Application No.PCT/JP2007/075356 with an international filing date of Jan. 4, 2008,which is incorporated in its entirety by reference herein.

FIELD OF THE INVENTION

The present invention relates to a method for displaying pieces ofinformation taken from divisional parts of a continuousthree-dimensional (3-D) face seamlessly onto a rectangular continuousplane, or displaying pieces of information on separate flat planesseamlessly onto a rectangular continuous plane. Typically, it relates toan information processing method especially addressed to reducing localdistortions among pieces of surface information of the earth displayedon a rectangular plane.

BACKGROUND OF THE INVENTION

“IPIX 360 Suite” is a commercial name of a commercially available systempresented by Minds-Eye-View Inc. (see http://www.ipix.com). This system,hereafter simply called “IPIX” in this application, deals with input andoutput of surface information of, in particular, spheres among various3-D objects. This system can generate spherical information by virtuallycoupling two circular fisheye images together. Users can get a view ofany partial portion of the spherical information by moving a rectangularviewer from one place to another.

Among various existing techniques for flattening out a spherical imageon a rectangular plane, cylindrical projections directed to producing aworld map can display the entire surface of the earth on a rectangularflat plane by projecting the earth's surface onto a cylindercircumscribing the equator and then unfolding the cylinder. Mercatorprojection is a conformal projection, and equal-area cylindricalprojection is an equal-area projection. Panorama photography is ananalogous art that captures images of some divisional parts of theentire view of 360 degrees and assembles these images on a virtualcylinder.

In 1879, Charles Sanders Peirce of the United States Coast and GeodeticSurvey proposed a quincuncial projection (see(http://www.progonos.com/furuti/index.html). This quincuncialprojection, hereafter called “Peirce's projection” in this application,can provide a square or 1:2 rectangular conformal world map and also cantessellate a plurality of such world maps on a plane. Geographicalinformation properly matches between adjacent world maps tessellated onthe plane.

As a scheme of polyhedral projection, Laurence P. Lee at NationalMapping Agency in New Zealand proposed in 1965 a conformal world map ofan equilateral triangle made by triangular projection that projects theearth's surface on a tetrahedron and then developing the tetrahedron(see http://www.progonos.com/furuti/index.html). This is hereaftercalled “Lee's Projection” in this application.

As a technique for correcting distortion of a flattened image convertedfrom spherical information, Buckminster Fuller presented a cartographycalled the Dymaxion map that has less distortion in continents' areasand shapes than any prior cartographies (see R. Buckminster Fuller,INVENTIONS: St. Martins' Press, 1983, P. 85). This cartography dividesthe entire spherical surface into twenty equal-area triangular regions,then projects information on respective spherical icosahedral regionsonto respective triangular face of a regular icosahedron, and thereafterdevelops the icosahedron. Therefore, each divisional spherical regioncorresponds to each icosahedral face, and the ratio of the area of eachdivisional triangular region relative to the entire area of the sphereis equal to the ratio of each icosahedral face relative to the totalsurface area of the icosahedron. That is, when the surface of the earthis projected on the icosahedron, the center angle of 63′25″ of each edgeof the divisional triangular region of the sphere is equally maintainedas the center angle of each edge of the icosahedral face.

Collignon's Projection by Edouard Collignon proposed an equal-area worldmap (see http://www.progonos.com/furuti/index.html) in 1865. Here isused a pseudocylindrical projection, which maintains an originalarrangement of longitudes radiating from the two poles and an originalparallel arrangement of latitudes on one rhombus or two rhombuses.

Commercially available software under the commercial name of “Flexify 2”(2008 Flaming Pear Software: http://www.flamingpear.com/flexify.html)utilizes map projections in image processing. It can convert importedflat images such as panorama or fisheye pictures into images on anytwo-dimensional form according to various kinds of map projections suchas Dymaxion map and the Peirce's projection, or into polyhedral images.

Japanese Patent Laid-open Publication No. JP 2003-178298 is related tocorrection of errors in area ratio, i.e. in solid angle, in photography.This application discloses a technique called “mesh camera” for pastingphotographs taken by different types of lenses to fit with each other.The “mesh camera” technique can project such photographs onto a 3-Dobject called an output frame such as, for example, a regular polyhedronor sphere, while minimizing the solid angle error.

U.S. Pat. No. 6,141,034 discloses a technique which can simultaneouslyexpose a substantially omnidirectional view by arranging externallyoriented optic axes on eleven dodecahedral faces. This patent proposesarrangements of optic axes based on a tetrahedron and an octahedron tocope with stereoscopic viewing.

All of the above-mentioned existing techniques are defective in oneaspect or another. Namely, the IPIX technology discussed above containslarge distortions when the image is zoomed out. The distortions becometoo large for a user to properly figure out the subjects when the imageis zoomed out to provide an approximately hemispherical visual field.Additionally, the IPIX is subjected to an undesirable phenomenon calledGimbal lock, which pertains to unnatural movements of the viewer, sincethe viewer rotates about a single axis.

Cylindrical projections leave distortions at bottom and top regions ofworld maps. Therefore, shapes of subjects in both polar regions aredifficult to figure out. These projections can rearrange geographicalinformation in the east and west directions, but cannot cope with suchrearrangement in other directions without a certain complex imageprocessing. Similarly, the panorama photography, which takes shots whilerotating a camera about a single rotation axis, cannot capture objectsabove and below the camera position. Therefore, the panorama photographyis not a technique for providing a complete omnidirectional image aswell.

The Dymaxion map that is just a development of an icosahedron inevitablyhas a zigzag outline, so that the geographical information in the worldmap is difficult to figure out properly. If ocean currents are added tothe world map, the currents appear discontinuous regardless of allefforts to rearrange the twenty regular triangles. Thus, the Dymaxionmap cannot pack geographical information closely on a rectangular planethat is an ideal map format.

Among regular polyhedra, the icosahedron employed in the Dymaxionprojection can divide an omnidirectional image into 20 equal parts andcan distribute distortions to 20 divisional regions equally. Incontrast, the tetrahedron employed by the Lee's projection divides anomnidirectional image into as much less as only four parts, and resultsin producing strong local distortions. Therefore, in the world map bythe Lee's projection, objects near the tetrahedral vertices increase insolid angle to five times or more of their original sizes. Additionally,the Lee's projection is not a proposal of a two-dimensional rectangularimage. Because of the same reason, the Peirce's projection is alsosubjected to a large distortion.

The Collignon's Projection is an equal-area projection with a simpleoutline. However, it does not provide a rectangular map.

The Flexify 2, which utilizes existing map projections, involves thesame problems as those of the Dymaxion map and the Peirce's projection.

JP 2003-178298 is not directed to flattening out omnidirectional images.Further, this prior art is not a proposal of a photographing technique.Instead, it relates to assembling some different shots taken bydifferent setups into an integral image. In this prior art, one or moreof prepared pictures are inevitably expanded locally when packed on asphere or a regular polyhedron, and this produces a resulting imagelocally different in resolution.

U.S. Pat. No. 6,141,034 does not teach any technique to flatten outomnidirectional images. In addition, this existing art cannot obtain acomplete omnidirectional view due to the lack of an optic axis towardthe bottom one of dodecahedral faces.

In summary, the existing arts of one group involve the problem that adistortion is contained when an entire area of an image taken from a 3-Dobject such as a sphere or a polyhedron is displayed simultaneously on arectangular flat plane; and those of another group involve the problemthat, once a distortion is corrected, the outline of image can no longerclosely fit in a rectangular plane of an ordinary display monitor, andproduces an unused zone called vignetting on the display.

In a further review of the entire field of omnidirectional photography,existing techniques step through some of various mapping processes suchas exposure, image processing and projection for display, respectively,from the stage of taking photographs up to the stage of giving aprojected image of the photographs to be observed on a screen. However,any of these existing arts deals with only a part of these processes,and none of them disclose a method of consistently stepping across allmapping processes.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide aninformation processing method capable of reducing local distortions inthe process of developing surface information of the earth onto a plane,or vice versa.

A further object of the present invention is to provide an informationprocessing method that can display continents and islands of the earthon a rectangular plane while maintaining their area ratio.

The present invention is characterized in generating information in aframe of a rectangular plane by repeating division or unification ofplanes typically by using inscribing or circumscribing sphericalpolyhedra. More generically, the present invention is characterized inmaintaining a constant area ratio between the total area and eachdivisional or unified plane. Thereby, the present invention can reduce,and/or equally distribute, the distortion of information.

According to a first aspect of the present invention, there is providedClaim 1

The present invention can thereby reduce, and/or equally distribute thedistortion.

The feature of the first aspect of the invention will be more easilyunderstood from FIG. 60 that shows a simple model using two circularfisheye photographs as information on the plurality of start faces.These two start faces SS1 and SS2 are adjacent to each other, sharingthe common point SP1. The start face SS1 is defined by a line SE1 andthe point SP1, and the start face SS2 is defined by a line SE2 and thesame point SP1. Each start face is deformed and packed to fill a partialzone of a rectangular plane exclusively assigned to it (RS1 and RS2). Inthis conversion, the positional relation between the two start faces SS1and SS2 is retained between the end faces RS1 and RS2 on the rectangularplane as well, and the lines SE1, SE2 and the common point SP1 definingthe original faces SS1, SS2 are preserved as lines RE1, RE2 and pointRP1 on the rectangular plane as well.

Still referring to FIG. 60, information at a location SS10 on the startplane SS1 corresponds to information at a location RS10 on the end planeRS1. That is, each piece of information on the start faces has aone-to-one relation with a piece of information on the end planesfilling the rectangular plane.

In the information processing method according to the first aspect ofthe invention, the information on the start planes is converted to theinformation on the end planes to fill the entire rectangular plane withno gaps or overlaps. Here is ensured that the first area ratio betweenthe total area of all start faces and the area of each start face isequal to the second area ratio between the total area of the rectangularplane and the area of each end face on the rectangular plane. This meansthat the following equation is substantially satisfied.

SS1/(SS1+SS2)=RS1/(RS1+RS2), SS2/(SS1+SS2)=RS2/(RS1+RS2)

Another more complex model is shown in FIG. 61. This is a development ofa polyhedron and its close packing onto a rectangular plane. Three linesPE12, PE13 and PE14 define a single unit start plane PS11 on a totalstart plane PS100. The unit start plane PS11 connects to a unit startplane PS12 sharing the line PE12, to a plane PS13 sharing the line PE13,and to a plane PS14 sharing the line PE14.

All unit (or divisional) start planes (faces) constituting the entirestart plane (face) PS100 are deformed and compiled to two-dimensionallyfill a rectangular end plane RS100 with no gaps and overlaps. LinesPE12, PE13 and PE14 defining the start unit face PS11 have beentransferred to lines RP12, RP13 and RP14 to define an end unit face RS11on the rectangular end plane RS100. Similarly, points PP12, PP13 andPP14 on the start unit face PS11 are retained as points RP12, RP13 andRP14 on the end unit face RS11. In this manner, lines defining all startunit faces are transferred to lines defining respective end unit faces,and points on all start unit faces are retained on respective end unitfaces. That is, information on each start unit face is in one-to-onerelation with information on each end unit face while keeping apositional relationship among respective start unit faces as arelationship among respective end unit faces as well.

The above two models deal with rectangulation (deformation onto arectangular plane) of two-dimensional information prepared beforehand byan existing flattening technique from an omnidirectional image.Occasionally, therefore, two faces that are continuous in the originalinformation may appear separated in a flattened form. Rectangulation bythe present invention can connect such separate faces to restore theiroriginally connected relation. This means that the invention improvesthe positional relation among unit faces by substantially recoveringtheir original relation beyond an intervening form.

The condition that the first area ratio of each start unit face relativeto the entire start face is substantially equal to the second area ratioof each end unit face relative to the entire rectangular plane can beexpressed by the following equation.

PS11/PS100=RS11/RS100

The first aspect of the present invention satisfies this condition fortwo-dimensional compilation of information on a rectangular plane.

In this respect, images two-dimensionally compiled images by existingcylindrical projections include inconsistency with original information.Two polar points are extended to lines sometimes as long as the equator.The Dymaxion map that uses polyhedron's development cannot pack theinformation on the polyhedral faces in a rectangular plane. The Peirce'sprojection and the Lee's projection, as well as the software “Flexify 2”applying these projections to cartographies, all fail to display the endinformation with the same area ratio as that of the start information.In contrast, the information processing method according to the firstaspect of the present invention can maintain the same area ratio as thatof the start information and the same relations of lines, points andplanes as those of the start information also in the end information.

When different faces of an object are caught as pictures by severalcameras, which are typically component cameras constituting a compositecamera, as shown in FIG. 2, planes of projection F26 taken by respectivecameras as illustrated by hatching in FIG. 62 are separate from eachother. These separate data can be imported from the different camerasand can be integrated to form adjacent faces F27 of a three-dimensional(3-D) object PG27. Therefore, in the context of the present invention, aplurality of adjacent faces each defined by one or more lines and one ormore points include not only those of original information but alsothose obtained as a result of certain processing of originalinformation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram showing the concept of a solid angle.

FIG. 2 is a diagram showing iso-area grids on a tetrahedron and aspherical tetrahedron according to the present invention.

FIG. 3 is an axonometric diagram of the two grids shown in FIG. 2.

FIG. 4 is a schematic diagram showing a grid generated by opticallyprojecting the spherical grid on a tetrahedron shown in FIG. 2.

FIG. 5 is an axonometric diagram of the two grids shown in FIG. 4.

FIG. 6 is a schematic diagram of grids subdividing the two grids shownin FIG. 2 according to the invention.

FIG. 7 is a schematic diagram of the earth mapped on a tetrahedron byiso-area mapping according to the invention.

FIG. 8 is a schematic diagram showing a world map made through theiso-area mapping process according to the invention and a comparativeworld map made by omitting the iso-area mapping process.

FIG. 9 is a schematic diagram of a tessellated image of some copies of aworld map with a viewer according to the invention.

FIG. 10 is a schematic diagram showing four world maps extracted andoutputted from the tessellated image shown in FIG. 9.

FIG. 11 is a schematic diagram showing another four world maps extractedand outputted from the tessellated image shown in FIG. 9.

FIG. 12 is a schematic diagram of a tessellated image shown in FIG. 9with another viewer framing a full-spherical field.

FIG. 13 is a schematic diagram showing world ocean currents of Januaryin form of a Dymaxion map.

FIG. 14 is a schematic diagram showing world ocean currents of Januaryin a world map according to the present invention.

FIG. 15 is a schematic diagram of iso-area grids on a cube and on aspherical hexahedron according to the invention.

FIG. 16 is a schematic diagram of a graticule iso-area grid onfull-spherical information according to the invention.

FIG. 17 is a schematic diagram corresponding to a top view of thefull-spherical information shown in FIG. 16.

FIG. 18 is a schematic diagram showing a graticule iso-area grid on anoctahedron according to the invention.

FIG. 19 is a schematic diagram showing two-dimensional developments ofthe grids shown in FIGS. 16 and 18.

FIG. 20 is a schematic diagram showing a grid on two opposite squarefaces on which an octahedral image has been mapped according to theinvention.

FIG. 21 is a diagram showing a process of integrating images mapped ontwo opposite faces and covering a full-spherical field onto a singlesquare plane according to the invention.

FIG. 22 shows three different images of a world map all flattened froman octahedral image according to the invention.

FIG. 23 is a diagram showing a result of tessellation of one of thethree images of the world map shown in FIG. 22 according to theinvention.

FIG. 24 is a diagram showing a result of tessellation of another of thethree images of the world map shown in FIG. 22 according to theinvention.

FIG. 25 is a diagram showing a result of tessellation of the rest of thethree images of the world map shown in FIG. 22 according to theinvention.

FIG. 26 is a schematic diagram showing iso-area mapping of an image froman icosahedron to a dodecahedron according to the invention.

FIG. 27 is a schematic diagram showing iso-area mapping of an image froma dodecahedron to a cube according to the invention.

FIG. 28 is a schematic diagram of a tessellated image showing an entiresequence of a scene according to the invention.

FIG. 29 is a schematic diagram of an image tessellated by a negativetemplate image according to the invention.

FIG. 30 is a schematic diagram of an image made by integrating theimages shown in FIGS. 28 and 29 for monitoring purposes according to theinvention.

FIG. 31 is a schematic diagram showing an aspect of optical projectionby a cubic omnidirectional camera according to the invention.

FIG. 32 shows a schematic diagram showing an omnidirectional camera witha concentric optical center according to the invention.

FIG. 33 is a cross-sectional schematic diagram for explaining the camerashown in FIG. 32 in detail.

FIG. 34 is a schematic diagram of a rectangular operation interface fortexture mapping according to the invention.

FIG. 35 is a schematic diagram showing a process of mapping atwo-dimensional image on a tetrahedron and a sphere according to theinvention.

FIG. 36 is a cross-sectional schematic diagram showing a process ofmapping a spherical image onto a medium of an arbitrary shape accordingto the invention.

FIG. 37 is a cross-sectional schematic diagram showing a process ofmapping an object of an arbitrary shape onto a sphere according to theinvention.

FIG. 38 is a schematic diagram showing a first half process ofmultilayer mapping of an arbitrary shape on an octahedron according tothe invention.

FIG. 39 is a schematic diagram showing a second half process ofmultilayer mapping of an arbitrary shape on an octahedron according tothe invention.

FIG. 40 is a schematic diagram showing a rectangular operation interfacefor a dome theater.

FIG. 41 is a schematic diagram showing a cubic composite camera with athree dimensional arrangement of planes of projection for each opticaxis according to the invention.

FIG. 42 is a schematic diagram showing a tetrahedral composite camerawith polyhedral surfaces of projection according to the invention.

FIG. 43 is a schematic diagram showing preferable shapes andarrangements of image sensors for subdivided iso-area grids according tothe invention.

FIG. 44 is a schematic diagram showing an octahedral composite camerawith three squares integrating a three dimensional arrangement of planesof projection for each optic axis according to the invention.

FIG. 45 is a schematic diagram of a cuboctahedral composite camera witha reflector suitable for capturing positional relations among camerasaccording to the invention.

FIG. 46 is a schematic diagram of a rhombicuboctahedral composite camerahaving diversity in arrangement of optic axes.

FIG. 47 is a schematic diagram of a composite camera available forrequirements of changes in resolution and/or stereoscopic imagingaccording to the invention.

FIG. 48 is a schematic diagram showing a method of obtaining astereoscopic image and a method of obtaining an omnidirectionalphotograph by the camera shown in FIG. 47 according to the invention.

FIG. 49 is a schematic diagram showing a method of obtaining anomnidirectional stereoscopic photo with a cubic composite cameraaccording to the invention.

FIG. 50 is a schematic diagram showing a combination of three cameras asan example of the composite camera shown in FIG. 47 according to theinvention.

FIG. 51 is a schematic diagram illustrating a process of mapping apolyhedral image on a rectangular plane via a tetrahedron according tothe invention.

FIG. 52 is a schematic diagram showing a process of quickly creating arectangular image by using two cameras according to the invention.

FIG. 53 is a schematic diagram illustrating a process of a hybridmapping method for open surfaces according to the invention.

FIG. 54 is a schematic diagram showing an eclectic method combiningcylindrical projection and rectangulation according to the invention byusing a rhombic dodecahedron as an example.

FIG. 55 is a schematic diagram showing iso-area division of a curvedsurface as a result of combination of a polyhedron and a plane by usingan octahedron as an example.

FIG. 56 is a schematic diagram of a camera suitable for a compactdesign.

FIG. 57 is a schematic diagram of a rhombic dodecahedral image and anoctahedral image captured by the camera shown in FIG. 41 according tothe invention.

FIG. 58 is a schematic diagram showing a method of dividing an objectwith curved lines according to the invention by using a solid havingarcuate meridians as an example.

FIG. 59 is a schematic diagram illustrating the method of dividing anobject with curved lines according to the invention by using quadraticcurves as an example.

FIG. 60 is a schematic diagram illustrating a model of the mappingconcept according to the invention for fisheye images.

FIG. 61 is a schematic diagram illustrating another model of the mappingconcept according to the invention for development of an image from apolygon.

FIG. 62 is a schematic diagram illustrating a process of uniting imageson discontinuous planes of projection onto one object according to theinvention.

FIG. 63 is a schematic diagram illustrating another model of the mappingconcept according to the invention for circular images.

FIG. 64 is a schematic diagram showing the mapping concept according tothe invention in comparison with an existing cylindrical projection.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the present invention are explained below in detail withreference to the drawings.

First Embodiment Tetrahedral Iso-Area Division

First explained is area ratio of a given face relative to the entirearea of a plurality of continuous faces is explained, taking a sphere asone of three-dimensional (3-D) objects as an example. Area ratio of agiven spherical area relative to the entire surface of a sphere iscalled solid angle. With reference to FIG. 1, solid angle is a valuerepresenting the extent of an object 2 as viewed from the center O1 of aunit sphere S1. Steradian (abbreviated “sr”) is normally used as a unitof solid angles. An area 3 on the unit sphere S1 (radius: 1) representsthe dimension of the object 2. This area 3 is a cross-sectional area ofa cone defined by rays of half lines extending from the center O1 towardthe outline of the object 2. The cross section is taken by cutting thecone along the spherical surface of the unit sphere S1. Maximum solidangle covering an entire view is 4π sr and a hemispherical visual fieldis 2π sr. In case of a world map, areas of continents or islands can bequantified by solid angles.

Numerical value of a given field of view by angle of view can beexpressed only as a lesser circle C2 on a unit sphere. Such a lessercircle is called an image circle. The image circle scheme can divide anomnidirectional view equally only when dividing it into two hemispheres.The existing IPIX technique discussed above relies on this scheme byusing fisheye lenses. In contrast, solid angles can describe any visualfields of any shapes in numerical values of their areas, and make iteasy to patch a plurality of images different in shape and size on asphere.

Mapping, including the technique using solid angles, which preserves thearea ratio of a given area on a solid's surface relative to the entiresurface area of the solid is hereafter called iso-area mapping forsimplicity. The “mapping” includes operations of optical projection,reflection, refraction, especially exposure for photography, projectionby a projector, and any operations of transforming a face, ofintegrating several continuous faces into one face, or dividing one faceto several continuous faces by transformation. Users can getquantitative comprehension of objects, which is difficult for humaneyes, by observing and diagnosing rectangular images made by Iso-areamapping. Iso-area mapping can be used for various purposes such asendoscopic diagnosis, cartography, measurement of magnetic forces andcalculation of sky factor (daylight factor), among others, to exactlydetermine sizes and distributions of seats of diseases, sizes of ozoneholes, magnetic flux densities and distributions, or projective areas ofbuildings.

For the purpose of rectangulation of an omnidirectional or partialimage, the use of a regular polyhedron makes it possible to equallydivide a complete visual field and evenly distributes face angles andinterior angles made by segments to the polyhedral faces upon mappingthe image on the polyhedron. FIGS. 2 and 3 illustrate iso-area mappingand rectangulation of an omnidirectional image via a tetrahedron. FIGS.2 and 3 show a grid G1 on a regular tetrahedron P10 and a grid G2 on aspherical regular tetrahedron that is tangential to the tetrahedron P10.FIGS. 4 and 5 show a grid G3 on a regular tetrahedron on which the gridG2 is projected from the optical center O10.

With reference to FIG. 2, a grid G1 subdivides each tetrahedral faceinto 24 congruent triangles (including mirror images) withperpendiculars to tetrahedral edges and lines parallel to theperpendiculars connecting points dividing the edges into quarters. Thisdivision results in uniforming different lengths of segments anddifferent interior angles of the grid G1 to less varieties. The hatchedregion 34 is one of the 24 triangles. Consequently, the grid G1subdivides all surfaces of the tetrahedron into 96 regions in total,which are equal in area.

Still referring to FIG. 2, the grid G2 subdivides a sphericaltetrahedral face into 24 triangles that are equal in area in thefollowing process. Great arcs that are geodesic lines connect midpoints30, 35 and 36 of edges of the spherical tetrahedron and its vertices V6,V7 and V5 respectively. The great arcs cross at a point O9. Points O9,35, V6 and 36 outline one spherical square. This spherical square isequally subdivided into eight spherical triangles by other great circlesthat cross at a point 28 and connecting vertices and midpoints of edgesof this spherical square. Here are contained two types of triangles(including mirror images). The other five spherical squares are alsosubdivided in the same manner and the sphere is divided into 96 regionsin total, which are equal in area.

On the other hand, a grid G3 is obtained by optically projecting thegrid G2 on a regular tetrahedron as shown in FIG. 5. Each node in thegrid G3 is on a segment connecting a node in the grid G2 and the centerof a concentric sphere. As shown in FIG. 4, two hatched sphericaltriangles 37 and 70 a that are equal in solid angle (area ratio relativeto the entire sphere) on the grid G2 are projected onto regions 37 a and70 b in the grid G3 respectively. As a result of this opticalprojection, the area ratio of the region 37 a relative to the entiretetrahedral surface becomes 5.1 times the area ratio of the region 70 beven though the regions 37 and 70 a were equal in area ratio before theprojection. It is an essential feature of iso-area mapping to maintainthe area ratio, i.e. solid angle, upon mapping the region 37 to theregion 34 on the grid G1 as illustrated in FIG. 2 in which the arearatio of the spherical triangle 37 on the grid G2 equals the area ratioof triangle 34 on the grid G1. This division implementing iso-areamapping is hereafter called iso-area division, and the dividing gridsensuring iso-area mapping between both surfaces before and afteriso-area mapping or between a pair of two corresponding grids arehereafter called iso-area grids.

Iso-area division need not use geodesic lines such as a great circle,and instead may use a lesser circle, other curves and/or a chain ofthese lines. The division may be partly uneven. The mapping methodaccording to the embodiment of the invention is applicable not only tospheres and tetrahedrons as illustrated but also to any other 3-Dobjects such as other regular and irregular polyhedrons, hyperboloidsand other open surfaces, objects having curved surfaces of surfaces ofrevolution, and so on. The method may use only a part of any of those3-D objects, for example, in form of a hemisphere suitable for measuringthe sky factor. The iso-area division includes a division usingtetrahedral edges alone. However, edges of polyhedrons other thanregular tetrahedron are better for use in applications like securitycameras and the like since a tetrahedron having only four faces islimitative in effect of distortion correction.

The iso-area mapping can cope with a desired frequency of division toupgrade its accuracy. FIG. 6 shows an eight-frequency iso-area grid G5of a spherical tetrahedron obtained by subdividing the grid G2 shown inFIG. 2. Based on this grid, spherical information is mapped on aneight-frequency iso-area grid G4 obtained by subdividing the grid G1shown in FIG. 2. For example, a spherical triangle 71 is mapped on atriangle 71. The explanation made above with reference to the firstembodiment is similarly applicable to all other embodiments as well.

FIG. 7 shows a tetrahedral image of the earth by iso-area mapping. Arectangular image is obtained by developing the tetrahedron along cutlines on a tetrahedral edge E3 extending between vertices 15, 16 andsegments E4 and E5 that connect the midpoint 9 of the edge E3 to othervertices 10 and 11 respectively.

In FIG. 8, SC1 denotes a rectangular omnidirectional image, which is anequal-area world map in this case, having the aspect ratio of 1:√3(L1:L2). Points 9, 10 and 15 shown in FIG. 7 have been transferred topoints 17, 20 and 18 in FIG. 8 respectively. The vertex 11 appearing inFIG. 7 has been transferred to a midpoint 19 in FIG. 8. Angles aroundthe vertex 11 have been developed to half surround the point 19 and makea straight angle since the sum of interior angles around a tetrahedralvertex 11 is 180 degrees. The method utilizes the geometrical nature ofa regular tetrahedron that can be unfolded into a rectangular plane.FIG. 8 also shows a world map SC1 a made by the same process exceptingiso-area mapping. The map equally distributes four tetrahedral verticeson a rectangle, and symmetry of the regular tetrahedron contributes tomaintain original of the solid angle of each face and the central angleof each edge. Therefore, local concentration of distortion does notoccur. Nevertheless, some figures near vertices are distorted to anincrease size.

The mapping process including multiple mapping steps such as the step ofmapping onto a spherical polyhedron with 96 faces and the step ofmapping onto a tetrahedron as explained with reference to the foregoingembodiment is hereafter called “multi-step mapping”. Iso-area divisionand iso-area mapping are not limited to multi-step mapping. Instead, anomnidirectional view may be mapped directly on a rectangular plane byiso-area mapping. Users can customize a frequency of division and/or adividing pattern of iso-area division as well.

For easier understanding of the present invention, explanation madeherein is based on geometry, using various geometrical terms likeregular triangle, congruence, etc. that have definite meanings. Uponpracticing the invention, however, inevitable errors or modificationsmust be allowed in actual operations of computers or in an actualprocess for making computer programs or physical products such ascameras. Therefore, note that the invention assumes such geometricalimperfections. In particular, the first embodiment allows anapproximation for practicing “iso-area mapping” or standardization inline length or interior angle provided it ensure the effects explainedwith reference to the first embodiment. For the same reason, theinvention allows minor lacks and/or deformation in mapped planes and/or3-D objects to be mapped. The remarks made in this paragraph areapplicable not only to the first embodiment but also to all otherembodiments.

In the foregoing explanation, the sphere inscribes to the regulartetrahedron. However, objects involved in the mapping, i.e. an object asa sender of information to be mapped (hereafter called a sender objectas well) and an object as a receiver of the information to be mapped(hereafter called a receiver object as well) need not have an inscribingor circumscribing relation. The sender object and the receiver objectmay be related such that, for example, each flat face of a polygon asthe receiver object intersects the spherical surface of a sphere as thesender object, or vice versa. In this case, local distortions betweenthe start image and the end image can be minimized. It is alsoacceptable that the sender object and the receiver object areconcentrically separate. The tetrahedron may be developed into arectangular plane by cutting along arbitrary segments thereon. Insteadof a rectangle, users can develop a tetrahedron to any other polygonsuch as a regular triangle, a plane outlined with some curves such as acircle, or a concaved face or any other arbitrary 3-D surface. Althoughtetrahedrons are suitable for obtaining developments, other polyhedronsare usable for making developments as well if other two-dimensionalgeometries are acceptable as developments.

The image of FIG. 9 is a result of tessellation of the obtainedomnidirectional rectangular image SC1. Broken lines E7 are mappedsegments of tetrahedral edges, and they make a three-way grid. From theimage of FIG. 9, some rectangular frames VR1, VR3, VR4 each covering oneworld map different in aspect ratio from the above-mentioned world mapSC1 can be obtained. These frames can change the direction and can slidein three directions 23, 24, and 25 along the three ways of the three-waygrid. Aspect ratio of these frames is L3:L4=4:√3.

SC1 is a world map locating Antarctica and Australia at a centralposition. By sliding the frame covering one entire world map, anyappropriate equal-area world maps that are secondary omnidirectionalviews can be provided. FIG. 10 and FIG. 11 show world maps with theaspect ratio of 4:√3 carefully trimmed from the tessellated image tocenter different regions and not to interrupt most of continents andislands. These world maps LC1 through LC8 are equal-area world mapscentering Middle East, Hawaii, Antarctica, India and China, U.K.,Mexico, Australia and Japan, respectively, in this order.

FIG. 12 is the same tessellation image as that of FIG. 9, and shows thatworld maps SC2, SC3 and SC4 having the aspect ratio of 1:√3 can beextracted in addition to the former world maps. In addition, a world mapwith a larger frame SC4 can be extracted for a better view of ageographical relation around regions (near Tasmania) at four corners ofthe world map, for example, SC4. Furthermore, a world map extracted witha frame SC400 covering 16π sr makes it easier to recognize geographicalrelations to anywhere in the world even from the regions at four cornersof the world map. These frames are suitable for continuously displayingworld ocean currents, sea and air routes, weather patterns, long termtracks of satellites, and so on, without interruption.

As compared with the world ocean currents of January shown in theDymaxion Map of FIG. 13, a world map according to the present inventionas shown in FIG. 14 shows the same ocean currents clearly, continuously.

The frame for extracting a part of the tessellated image need not berectangular. Instead, any other polygonal outline such as a triangularoutline may be used. For example, only a part of the world such as thenorthern hemisphere may be extracted. Although the method has beenexplained by way of tessellation of a tetrahedral development in thelengthwise and crosswise directions, it contemplates other polyhedraldevelopment as well. Besides, the tessellation can be with gaps and/oroverlaps as long as it contains some connections with seamlesscontinuation. The tessellation can be arranged along any format such asa linear or a circular format. Tessellated images with localdiscontinuities are also contemplated by the present invention.

The first embodiment has been explained by way of a world map made byviewing the earth, taken as an object, from every external directionstoward its center. However, irrespectively of optical axes beingoriented inward or outward, omnidirectional photographs taken by acamera outwardly oriented from an internal single viewpoint can betreated as spherical images as well. These schemes of mapping have beenregarded as different technical issues in existing techniques, but thepresent invention deals with both these schemes equally. Therefore, thetechnical matters explained here are applicable to other embodiments aswell, and the first embodiment explained above may be used inomnidirectional photography as well.

Second Embodiment

Iso-area division of a cube is next explained as a second embodiment ofthe invention. A cube can be readily transformed to a tetrahedron byconverting four of eight cubic vertices to four faces. Therefore,information on a cube can be clearly remapped on cubic faces byphotographing setting six optical axes along the cube. FIG. 15 shows aset of a cubic iso-area grid G6 and an iso-area grid G7 of a sphericalcube circumscribing the iso-area grid G7, sharing common vertices. Thegrid G6 is an iso-area grid made by lines connecting diagonal corners ofeach cubic face and lines connecting mid points of all segments toequally divide each cubic face into sixteen triangles. One of thesetriangles is a region 82. In this manner, a cubic surface is divided tocongruent 96 triangles.

In the grid G7, diagonal corners V8 a, V9 a, V10 a, V11 a of a sphericalcubic face are connected by arcs (great circles), and their intersectionO13 a and midpoints of cubic edges are connected by arcs. Further,midpoints 78 through 81 of the great circles between the intersection 13a and the respective corners V8 a, V9 a, V10 a, V11 a are connected tothe midpoints of the cubic edges. Thus, the cubic face is divided tosixteen equal spherical triangles. One of these sixteen sphericaltriangles is the region 82 a. By similarly dividing the other threecubic face, the spherical face (spherical cube) is divided to 96 equalparts. In this manner, the region 82 a on the sphere can be iso-areamapped at a region 82 on the cube. If a cubic image obtained by theiso-area mapping is iso-area-mapped on faces of a regular tetrahedronhaving the points V8, V9, V12, and V13 at its apexes, a regulartetrahedron having an iso-area-mapped image is obtained. If the image onthe spherical cube is similarly mapped on a tetrahedron having pointsV10, V11, V14, V15 at its apexes, another regular tetrahedral image isobtained.

Third Embodiment

The iso-area division according to the invention includes a method ofequally dividing a solid angle and next subdividing the divided solidangle with a network system such as graticule. It further includes amethod of iso-area-mapping information on a polyhedron having arectangular cross section onto the rectangular cross-section, therebyflattening out the image information.

With reference to FIG. 16, the earth's surface, taken as typicalomnidirectional spherical information, is equally divided along edges90, 91, 92 of a spherical regular octahedron. Numeral 90 denotes theequator. Numerals 91 and 92 denote longitudes. A grid G12 subdivides thespherical regular octahedron with its graticule. One of the divisionalregion is shown at 89. A part of the grid G12 is shown on a sphere S6.It is preferable that, in this arrangement, the longitudes make aconstant angle between every adjacent ones and the latitudes align inparallel to each other. Intersections 93, 94 of the longitudes 91, 92correspond to the north and south poles in this case. FIG. 17 is a viewcentering at the North Point.

FIG. 18 shows a part of an iso-area grid G9 on an octahedron P5 sharingvertices with and inscribing to the spherical octahedron S6. Threesquares 95, 96 and 97 share vertices of the octahedron as theirvertices. The grid G9 comprises longitudes, which connect a point 93(for example, north pole) to points 102, 103, 104 equally dividing anoctahedral edge, and latitudes aligned in parallel to the segmentconnecting the vertices 98, 99, In this grid G9, the longitudes may makea constant angle between every two adjacent ones. A region 89 a is oneof divisional regions made by the grid G9. In this manner, respectiveregions are iso-area-mapped from the grid G12 to the grid G9, and theregion 89, for example, is mapped on a region 89 a.

A grid G11 shown in FIG. 19 is a development of the grid G12 shown inFIG. 12. A grid G9 is a front elevation of the grid on the regularoctahedron P5 shown in FIG. 18. To ensure iso-area mapping of eachdivisional region from the grid G11 to the grid G9, distances betweenlines corresponding to latitudes are adjusted to satisfyh1:h2:h3:h4=22.3:21.5:19.9:17.5.

After the same mapping operation with the grids is carried out on allother octahedral faces, a complete octahedral image P5 obtained therebyis next iso-area-mapped on front and back faces (F4 and F5) of one ofthree squares 95-97. FIG. 20 shows a layout of these front and backfaces F4 and F5 of the square region 95 both now having mapped images.The grid G9 on the octahedron has been mapped on a grid G10, and theregion 89 a has been mapped on a region 89 b. This side-by-sidealignment of the layout of the regions F4, F5 as illustrated in FIG. 20turns out a rectangular image of the aspect ratio of 1:2. FIG. 21 is anaxonometric diagram of the octahedron P5 shown in FIG. 18. By dividingthe back face F5 into four triangles with its diagonals and integratingthem, turned upside down, with the front face F4, a squareomnidirectional image having four corners 106, 107, 108, 109 isobtained. The operation carried out by using the square 95 can bereplaced with the same operation using, instead, the square 96 or 97.Thus, three rectangular images can be made in total. This method capableof creating three different square images from a single omnidirectionalimage is significantly advantageous as compared with any existing art.These grids G12, G9, G10 are hereafter called “graticule-iso-areagrids”.

For easier understanding of the present invention, explanation madeherein is based on geometry, using various geometrical terms that havedefinite meanings. Upon practicing the invention, however, inevitableerrors or modifications must be allowed in actual operations ofcomputers or in an actual process for making computer programs orphysical objects such as cameras. Therefore, note that the inventionassumes such geometrical imperfections. The terms, “equal”, “parallel”and “ratio of 1:2”, should not be construed in their strict definition.Instead, such terms herein contemplate a reasonably loose extent ofapproximation to involve the meanings of “substantially equal”,“substantially parallel”, and “ratio of approximately 1:2”.

The same mapping method is applicable to any 3-D object having arectangular cross sections each defined by two points on a face of theobject and one or more points on another face of the object.

World maps LC13, LC14 and LC15 shown in FIG. 22 are examples of imagesthat can be obtained by the above-explained method. These maps keep anequal-area layout even for the Antarctica and the icy field in theArctic Ocean that are distorted in the Mercator projection, and maintainthe latitudes concentrically parallel and longitudes radially extendingfrom the poles.

In this and other embodiments, the grid used for the division may bedisplayed. Scales of longitudes, latitudes, angles of elevation, anglesof direction, distances, etc. may be indicated as well. It is alsopossible to omit a mapping process on an octahedron and directly map aplane image such as a world map by an existing cartography, a panoramaor a fisheye image on a rectangular plane while eliminating distortions.

FIGS. 23 and 24 show images made by tessellating the world maps LC14 andLC15 respectively as unit or cell images. It may sometimes occur that asubject such as the Antarctica or an icy field on the Arctic Oceanappearing near a midpoint 140 or 141 of an edge of the square of oneunit image connects to the same subject of a neighboring unit imagealong their border. FIG. 25 shows an image made by tessellating theworld map LC13. In this map, all continents are displayed properly. Assuch, this method according to the invention presents threeomnidirectional images and permits the best one of them for a particularpurpose to be selected.

Further, a user can extract a secondary square unit LC20 (world map)defined by four corner points 122, 133, 134 and 132 centering at theNorth pole from a tessellated image shown in FIG. 25, as well as a worldmap as a secondary rectangular unit LC16 having the aspect ratio of 1:2defined by four corner points 121, 129, 130 and 131. These maps displayboth the arctic and Antarctic polar regions accurately, and presenteasy-to-view images of the southern hemisphere that is criticallydistorted in a map by the Lambert's azimuthal equal-area projection.From this tessellation, users can extract a world map as a secondaryrectangular unit LC17 having the aspect ratio of 1:4 as well.

The secondary unit LC17 functions as a viewer that can rotate and slidethe viewer frame in the directions pointed by arrows 113. By moving theviewer frame, the viewer can extract a rectangular world map LC18. Theworld map LC18 functions as a viewer too, which can rotate and slide inthe directions pointed by arrows 114. The secondary unit LC16 functionsas a viewer too, which can rotate and slide in the directions pointed byarrows 115. By moving it, the viewer can extract a rectangular world mapLC19 as well. The world map LC19 functions as a viewer too, which canrotate and slide in the directions pointed by arrows 116. As such, thismethod permits a user to select a viewer of a desired aspect ratiocovering up to an entire view in maximum to obtain the best frame of aworld map centering at any desired area of the world.

In this and other embodiments of the invention, the viewers may keeptheir rectangular shapes during zooming, sliding and rotating, or theviewers may change their aspect ratios and/or outlines to any arbitraryshapes. The iso-area division may be simplified and/or approximated,taking account of a limited speed of computer calculation though thedistortion correction becomes limitative. Further, the process ofsubtracting information of distance for importing a view into a unitsphere may be omitted. In this case, the process will result inmodifying a 3-D space. For example, if information of the universe withstars and other astronomical information is converted while keepinginformation of distances, observers can command, from a certainviewpoint (such as the earth), an entire visual field of night sky withstars in a rectangular omnidirectional image, and can realize, fromanother viewpoint, a particular 3-D space in which they can figure outdistances to respective celestial bodies.

Another method for measuring distance on a two-dimensional display isprovided by zoning a space of a visual field by a series of concentricspheres with different radii. Subjects in each divisional zone betweentwo spheres are projected on one of the two spheres and thenrectangulated according to the invention. Each rectangle displays onlysubjects located within the distance between two radii. Thereafter, aseries of rectangle images are tessellated in the order of lengths ofradii. As a result, the obtained tessellation sorts out the subjects inthe order of their distances. A user can figure out a distance of anyobject described as a grid value of a 2-D coordinate. These mattersexplained with reference to the third embodiment are applicable to allother embodiments as well.

Fourth Embodiment

The above-explained method may be modified to replace an octahedron witha tetrahedron. FIG. 51 schematically shows a tetrahedral omnidirectionalimage PG16. Two tetrahedral triangles are mapped on a front side of asquare F23 and two other triangles are mapped on a back face of thesquare F23. An omnidirectional rectangular image is obtained by flippingthe one side of the square F23 and uniting it to the other side. Thismethod is usable as long as the square F23 is approximately parallel toa tetrahedral edge regardless of whether the square locates in or out ofthe tetrahedron PG16. The obtained image can be tessellated in the samemanner as explained with reference to preceding embodiments.

Further, a couple of hexagonal images is obtained by diagonally mappinga cubic image on two sides of a hexagonal section. The obtained imagecan be tessellated with no gap and with some connections with seamlesscontinuation. The method can be also applied to any polyhedron as longas a link of its edges composes a polygon along a great circle. Forexample, a couple of decagonal images is obtained by mapping anicosi-dodecahedral image on a decagonal section. A plurality of theobtained images can be tessellated with some gaps and with someconnections with seamless continuation. The method can map an image ononly one side instead of two sides. A tetrahedral image may be re-mappedon an octahedron PG17 to flatten out an image according to the methodintroduced above.

Fifth Embodiment

The method according to the invention includes a tessellation methodviewing an entire event for certain duration, hereafter called“chronological tessellation” as well. FIG. 28 schematically shows achronological tessellation MX1 composed of 60 (vertical)×60 (horizontal)rectangular omnidirectional image units SC20, photographed at intervalsof one second and tessellated chronologically. The image unit SC20 takenhere is an image of an interior space with stairs and furniture. Withthis arrangement, it displays entire events for one hour at once withtwo different time scales: seconds in the horizontal direction andminutes in the vertical direction. With a chronological tessellation,spatial and sequential information of adjacent units properly matchesseamlessly. It can be displayed as an animation by viewing the unitimage SC20 one after another in chronological order. In this aspect, achronological tessellation is a video format. Since the chronologicaltessellation shapes a rectangle, it can record an entire animation injpeg or any other wide-use format for a still image. Time frames are notseparated by visible boundaries and maintain seamless continuation ofstill objects across the frame boundaries.

Each unit SC20 may display an animation instead of a still image. Ifeach unit SC20 shows an animation of one-second duration, a user canobserve the entire sequence in detail in one second. Twenty-five imagestaken in a second may be overlaid on a region of unit SC20 or arrangedthree dimensionally in depth. This method can be customized to clearlyshow a subject for various needs by adjusting the exposure time,interval of exposure and number of columns and rows of tessellation. Anobserver can sweep all events of a full day by a chronologicaltessellation composed of 24 (vertically)×60 (horizontally) unitsphotographed every one minute and tessellated in the chronologicalorder. In case of monitoring a crowded station, a chronologicaltessellation of unit images photographed by the shutter speed of oneminute filters out moving passengers and extracts only things long leftin the crowd such as suspicious packages. A chronological tessellationneed not arrange unit images chronologically. If it is more efficient toobserve, units or parts of units may be arranged in an unchronologicalorder. The technical matters explained in this paragraph are applicableto all other embodiments as well.

The invention includes a monitoring method for extracting only unusualevents. FIG. 29 schematically shows a tessellation image MX2 generatedby tessellating a unit image SC20 a repeatedly. The image SC20 a is aninverted negative image of SC20 and it is used as a template imageshowing a normal condition of the space to be monitored.

FIG. 30 schematically shows a monitoring display MX3 combining thetessellation images MX1 and MX2. White blanks in the display representnothing has been changed. It keeps the display surface blank unless anymovements or changes occur. A subject's negative image on MX2 perfectlymatches and cancels out its positive image, complementary colors, onMX1. Once a change or a movement occurs in a visual field, images on thetessellation images do not match and cannot cancel the colors. Anun-matched part emerges as a FIG. 61. This method permits a user to findan accident by observing the display MX3 and to start inspection of thetime frame around the FIG. 61 on the display MX3 by moving the viewer.

Usually, the most part of monitoring display MX3 remains white as longas an ordinary condition is maintained in the monitored space. Widelyused file formats such as jpeg can efficiently compress such images andcan reduce the file size when the data is saved in hard discs. With atemplate image SC20 saved separately, it is possible to restore anoriginal chronological tessellation image by combining the templateimage SC20 with the image MX3. This method is advantageous for long timevideo recording with a full frame in a full motion which requires a bigcapacity of data storage.

The template image SC20 a may be renewed to update the ordinary state inaccordance with changes of conditions such as daylight, for example. Atessellation image MX2 may contain several template images taken underdifferent conditions. Several template image units or parts of units canbe arranged in an unchronological order. It is also possible to modifythe tessellation image MX1 by delaying, which means shifting one or moretime frame for tessellating units and inverting the color, which cansubstitute for template tessellation MX2. In this case, this method canprovide a monitoring display MX3 by combining a chronologicaltessellation and the modified tessellation image. In this case, thedisplay notifies a change as a figure during the delay of time frames.It also can invert a chronological tessellation image instead oftemplate tessellation image. It also can obtain a monitoring display byimage enhancement and suppression by controlling each color componentRGB, CMYK, HSB or Lab) and/or opacity so as to cancel the color,brightness, saturation, and/or hue of unchanged object so as toemphasize the desired subjects.

Sixth Embodiment

The invention includes a method of iso-area-mapping an image from oneplane to another. FIG. 63 shows such a method of mapping a couple ofcircular images SS200 taken by a fisheye lens on a rectangular planeRS200. The images SS200 constitute an omnidirectional view taken by twotimes of exposure. Each of the circular images covers 2π sr visualfield. Here is made a grid composed of segments SE20 on the basis ofiso-area division by the grid G2 shown in FIG. 2, taking account of theprojection characteristics of the lens, which inevitably influence theprojected image, to ensure that an original solid angle of eachdivisional region be equal. The segments SE20 include curves. The gridcomposed of segments SE20 divides the region SS200 into 96 regions.

Still referring to FIG. 63, a grid composed of segments RE20 divides therectangular region RS200 into 96 regions equal in area (solid angle).Numeral RS20 denotes one of these divisional regions. Both the grid onthe circular images SS200 and the grid on the rectangular region RS200are generated according to iso-area division by the grid G2 shown inFIG. 2. Therefore, each divisional region SS20 in the region SS200 hasan exclusive correspondence with a particular one of the divisionalregions RS20 in the rectangular region RS200. Thus, circular imagesSS200 are iso-area-mapped on a rectangular plane by mapping individualregions SS20 on their respective counterpart regions RS20.

In this mapping process, it is necessary to adjust an area ratio of aregion when mapping on a rectangular plane to maintain its originalsolid angle when photographing.

Seventh Embodiment

The invention includes a method of mapping a plane image taken by anexisting technology, such as an existing world map, on a rectangularplane according to the invention while reducing errors in solid angle.In FIG. 64, a region CR10 represents an existing cylindrical projectionof a full spherical image SPR10 of, for example, the earth. SegmentsSPE1 through SPE8 are transferred to segments CE1 through CE8respectively. Regions SPR1-SPR8 are transferred to regions CR1-CR8respectively. Segments CE1-CE8 equally divide a region CR10 into eightregions CR1-CR8. On the other hand, a region OR10 is a rectangular planeregion based on the method shown in FIG. 20. Segments OE1-OE8 equallydivide a region OR10 into eight regions OR1-OR8.

Still referring to FIG. 64, the entire region CR10 is iso-area-mappedand rectangulated on the region OR10 by mapping the individual regionsCR1-CR8 to their respective counterpart regions OR1-OR8. SegmentsCE1-CE8 are mapped on segments OE1-OE8. A segment CE-N is originally apoint representing the North Pole and it is mapped on a point O-N tomaintain more faithful relative geographical relation on the sphericalsurface. Another segment CE-S is originally a point representing theSouth Pole, and it is mapped on four points O-S at corners of arectangular region OR10 to maintain more faithful geographical relationon the spherical surface. Although the south pole is divided to fourpoints O-S, this suggests that the world map can be rearranged to gatherthe four points O-S at one central point.

The region OR10 typically represents one entire world with no gaps andoverlaps. However, since the points O-S are divided to four cornerpoints of the region OR10 to display the information of one entire earthsurface of the earth without gaps and overlaps, it is difficult tofigure out the geographical relation around the points O-S. In the past,it had been misunderstood that the world is on an infinite plane sinceboth a sphere and an infinite plane do not have a dead end, and anypoint on the surface has expanses to all directions. In that sense,standard world maps developed on planes cannot represent perfectgeographical relations in peripheral regions.

In contrast, the method according to the invention can restore aninfinite plane in a two dimensional format by tessellating regionsOR1-CR8 repeatedly around a region OR10 to represent an endless expanseof spherical surface. Thus, the method according to the inventionmaintains geographical relations of any points or zones (such as theperipheral regions OR5-OR8) of the world to their neighbors in alldirections. For instance, the relation of a region SPR8 on a spherebordered on regions SPR4, SPR5 and SPR7 is represented on a hatchedregion OR8 bordered on regions OPR4, OPR5 and OPR7 on a two dimensionalformat.

Eighth Embodiment

Next explained is a method of obtaining a rectangular plane image viamulti-layer mapping (or multi-step mapping) using, especially, regularpolyhedrons. A mapping between regular polyhedrons can evenly distributeerrors of a solid angle, i.e. an area ratio, a central angle, a faceangle and an interior angle by a simple optical projection. FIG. 26illustrates a part of a mapping process from an icosahedron to adodecahedron. A dodecahedron is obtained by topologically convertingicosahedral vertices to a face and icosahedral face to vertices. Anicosahedral triangle is evenly divided into six congruent regions (incl.mirror image) by segments connecting the triangle's center 39 to itsvertices 38 and midpoints of its edges respectively. One of theseregions is the hatched region 72. An entire icosahedral image is evenlydivided into 120 regions by implementing the same process in all othericosahedral triangles. The icosahedral image is iso-area-mapped on aninscribed dodecahedron. A dodecahedral pentagon is equally divided intoten congruent regions (incl. mirror images) by segments connecting thepentagon's center 4 to its vertices 39 and midpoints of its edgesrespectively. One of these regions is another hatched region 73. Eachdivisional region such as a region 72 is mapped on each correspondingdivisional region such as a region 73.

FIG. 27 illustrates the next part of the mapping process from thedodecahedron on a cube. The dodecahedron inscribes the cube. An edge 41of the cube tangent to the dodecahedron divides a dodecahedral face intoa triangle 46 and a trapezoid 48. These regions are mapped on a triangle46 a sharing points 49 a, 42 and 43 on the cube as its vertices and on atrapezoid 48 a sharing points 49 a, 50 a, 44 and 43 on the cube as itsvertices. Positions of the points 49 a and 50 a on the cube arepreferably adjusted to ensure that the mapping is just an iso-areamapping. A resulting cubic image is again mapped on a tetrahedron or anoctahedron and thereafter flattened out to present a rectangular image.

The multi-layer mapping is applicable for any 3-D object and especiallysuitable for combinations of objects having symmetries and/orgeometrically common grounds. Multi-layer mapping by optical projectionfrom a geodesic sphere to any of a quasi-regular 32-face polyhedron(truncated icosahedron), regular dodecahedron and regular icosahedron isalso suitable as iso-area mapping because even a simple opticalprojection ensures iso-area mapping.

As such, the method according to the invention makes it possible toultimately rectangulate any image by combining operations of dividingand/or unifying faces of a polyhedron.

As mentioned above, for easier understanding of the present invention,explanation made herein is based on geometry, using various geometricalterms that have definite meanings. Upon practicing the invention,however, inevitable errors or modifications must be allowed in actualoperations of computers or in an actual process for making computerprograms or physical products such as cameras. Therefore, note that theinvention assumes such geometrical imperfections. For this reason, thegeometrical terms, “regular polyhedron”, “midpoint” and “equal division”used in this eighth embodiment and other embodiments of the inventionmay be interpreted to mean “polyhedron,” “point nearby midpoint” and“division” provided the same effects as those of the eighth embodimentare retained.

Ninth Embodiment

A reversed process of any of the mapping methods according to thepreceding embodiments is another embodiment of the present invention aswell. Especially, it is useful for texture mapping that is a techniquefor mapping a plane image on a 3-D object. In this texture mapping, anoperator has to inspect the mapped image from a variety of angles toconfirm whether all images are properly mapped on a desired object. Ifany visible seam or a distorted part is found, which degrades theappearance of the end image, the operator is involved in atime-consuming work to modify and align an image with others. Theinvention, however, provides a rectangular display that covers more thanone entire surface of a desired object, in which an operator can workson a texture image while checking seams on individual edges of an imageand can next iso-area-map the obtained image on a desired object withoutvisible distortions and seams.

FIG. 34 schematically shows a rectangular interface. A rectangularregion TM1 covers an entire surface field in which a user draws anentire surface image for a desired 3-D object. Broken lines represent apart of the same iso-area grid as illustrated in the first embodiment,which divides a part of the region TM1. A triangle 147 is one ofdivisional regions made by the grid. The other regions are divided aswell in the same manner. The image drawn in the region TM1 istessellated around the region TM1 to permit a look over the joint lineof adjacent ends of every two tessellated regions TM1. Here is used animage region TM2 larger than 4π sr. For example, the image in a region145 is displayed in a region 145 a as well.

FIGS. 35 and 36 show a reversed process of the first embodiment. Withreference to FIG. 35, a tetrahedral image obj4 is prepared by foldingthe rectangular image TM1 of FIG. 34. This tetrahedral image obj4 isiso-area-mapped into a spherical image obj5. A cross section taken alonga great circle 159 is shown in FIG. 36, which also shows a positionalrelation between the sphere of the spherical image obj5 and a desiredobject obj6. In this schematic diagram, the spherical image obj5 isprojected on the object obj6 with its optical center located at a center160 of the sphere of the image obj5 for easier explanation. If it isnecessary to implement an iso-area mapping through all processes, theprojection should be iso-area-mapping as well and does not have anoptical center. The object of the spherical image obj5 may becircumscribed to or intersect a subject to be mapped.

In this fashion, especially for an animation that exposes a subject frommany angles, an operator can carry out a texture mapping operation whilecommanding the substantially entire surface image of the subject to beseamlessly mapped. A rectangular image may be directly iso-area-mappedon a desired object without multi-layer mapping via a sphere or atetrahedron. The reversed image processing is applicable not only to thefirst embodiment but also to other related embodiments.

Tenth Embodiment

The invention includes a method of mapping an entire surface image of adesired object onto a rectangular plane possibly via a mapping on asphere. FIG. 37 shows a cross-sectional view of an object obj1 that is ahuman head. A spherical image 162 is obtained by dividing an entiresurface of the object obj1 into regions, next capturing each divisionalregion by setting an internal optic axis of each component camera fromall direction and then mapping these captured images on the sphere 162.The obtained spherical image is flattened out to generate a rectangularimage according to the invention. The object obj1 can be mapped directlyon a tetrahedron without a mapping process on a sphere.

Eleventh Embodiment

The invention includes an iso-area mapping and multi-layer-mappingapplied to arbitrary 3-D objects. FIGS. 38 and 39 schematically show amulti-layer-mapping process of a substantially entire surface of anarbitrary object on an octahedron. In FIG. 38 single dot and dash lines155 and 156 equally divide an entire surface of an object obj3 (in thisillustration, a head) into eight regions. One of the regions is ahatched region AR1. Geometry of the region AR1 and other regions can bedescribed by many polygons. In other words, the polygons compose manypyramids. One of them is a pyramid AR2. In its detailed illustration,triangles SR4, SR5 and SR6 compose pyramid surfaces of AR2 and aremapped on triangles SR4 a, SR5 a and SR6 a on its base triangle definedby vertices 157,158 and 171. Thus, the mapping unifies some polygons toone polygon by maintaining an area ratio of SR4:SR5:SR6 to be equal toSR4 a:SR5 a:SR6 a so as to be an iso-area mapping.

FIG. 39 shows an object obj2 obtained by applying the same operation toother regions. An octahedral image is obtained by repeating thisiso-area mapping for unifying polygons. Thus, many faces on a region AR1are integrated to an octahedral triangle. A desired image is obtained byagain unifying octahedron's two square pyramids to two sides of aninscribed square. The unification with a multi-layer-mapping graduallysimplifies geometry of a 3-D object, which can rectangulate a geometrythat contains a dead angle for a simple optical projection with a singleoptical center. The illustrations are modified for a better explanation.

Twelfth Embodiment

The invention includes a rectangular interface for image operation whilecommanding an entire visual field. A projection of an omnidirectionalimage on a spherical screen, such as a dome theatre needs severalcameras, typically component cameras of a composite camera unit, forinput and several projectors for output for dividing and sharing visualfields. An existing interface does not maintain solid angles of inputand/or output image, and causes distortion in a displayed image. Thismakes it difficult for an operator to figure out each divisional imageand its connection to an adjacent one. The method arranges individualdivisional images from respective cameras in a rectangular interfaceformat while maintaining solid angles of their visual field. Thus, themethod enables user to operate images for constructing anomnidirectional image while commanding an entire view with proper solidangle.

FIG. 40 shows a rectangular region TM3 utilizing a tetrahedraldevelopment which commands an entire visual field for sphericalprojection screen. A rectangular interface TM4 contains the region TM3to show if its seams connect adjacent regions properly. In case of anoperation to input a cubic image according to six cameras and to outputthe image to be displayed on a spherical screen by twelve projectorsarranged according to an dodecahedron, the method equally arranges anddisplays the six input images in six parallaxes such as a hatchedregions 163 which is defined by segments 165 and equally arranges anddisplays the twelve output images in twelve pentagons such as a hatchedregions 164 which is defined by segments 166.

Similarly to the arrangements, each hatched region 167,168 or 169represents a divisional image of an icosahedron, tetrahedron andoctahedron for input and output. Thus, the invention provides arectangular interface which commands more than an entire visual field inwhich operator can conveniently stitch each image and synchronize eachanimation while checking seams on every edges of the image to project itproperly on a screen. It is able to display divisions of angles ofelevation and direction. The method can be applied to a technologyrelating to a holography.

Thirteenth Embodiment

The mapping process in the invention includes an exposure that is anoptical projection. FIG. 31 schematically shows an exposing method toequally divide an omnidirectional image into six ⅔π sr visual field.Great circles 170 divide an omnidirectional image and compose aspherical cube S7. Each component camera 60 is installed at the centerO5 of the spherical cube S7 with its external optic axis AX1 towards thecenter O15 of the each cubic face PG1 tangent to the spherical cube S7.Angle of view of the camera is wider than 109.4712206 degrees to coverthe divisional ⅔π sr visual field. Thus, the captured images are mappedon a spherical cube to be rectangulated by the operation introduced inthe second embodiment.

Arcs connecting points 83, O14, 84, V8 b and 81 a on the spherical cubeS7 by great circles compose a part of the iso-area grid G7. It shows apart of grid G8 on a cubic face obtained by internal projection of thegrid G7 with an optical center O5. The grid G8 can be utilized as afilter. An obtained photograph by the camera 60 is iso-area-mapped onthe grid G6 shown in FIG. 15.

Fourteenth Embodiment

The invention relates to a mapping in which surfaces to map and to bemapped are separated, including a case of mapping (projecting) a view ona plane of projection via mapping (reflecting) on a reflector facing tothe plane of projection from a certain distance. The followingexplanation illustrates an example that simplifies operations to alignand unite divisional fields of view.

It is impossible to concentrate optical centers of a plurality ofcameras because of the existence of the camera bodies. A compositecamera unit having such component cameras has difficulty to take aclose-up picture. FIG. 32 schematically shows a method for asimultaneous exposure by four cameras at tetrahedral vertices with aconcentric optical center. Each camera 54 with an internal optic axistowards a reflector P12 captures a reflected image on the reflector P12.Four devices are arranged to cover an entire visual field while sharinga common virtual optical center O12. A curvature of the reflector P12are adjusted according to the angle of view of the camera and distancefrom the optic center O12. It is convenient for the reflector's geometryto correspond to an arrangement of cameras. A tetrahedral reflector P12fits to tetrahedral arrangement of the four cameras 54 as illustrated.

FIG. 33 shows two cross-sectional diagrams both taken along an opticaxis 56. One of these cross-sectional views, 143, shows a combination ofa camera 54-1 having a lens wider than 141° and a reflector. In a modelusing such a wide-angle lens, the reflector makes a flat reflection faceREF1. The other cross-sectional view, 144, shows a combination of acamera 54-2 having a typical angle of view and a reflector. In a modelusing such a typical lens, the reflector makes a reflection face REF2curved to present 1π sr visual field to the camera 54-2. A defect of themethod may be that the obtained entire image captures the camerasthemselves too. If it is disturbing, another camera 55 with an externalaxis 56 can be attached behind the camera 54-1 or 54-2 to replace theimage of the camera 54-1 or 54-2 with an image captured by the camera55.

Fifteenth Embodiment

The invention includes a multi-layer-mapping via a series of 3-Dobjects. Meanwhile the invention includes a photographing method toquickly obtain a rectangular image by reflector. It reduces compleximage processing after exposing and is able to display a sequence ofreal time animation without delaying its time frame. FIG. 52 shows acomposite camera unit PG18 composed of two component cameras 225 withtwo square-pyramid-shaped reflectors REF3 each facing to a counterpartone of the cameras 225. The two reflector shares a square base F18 ofthe pyramids and a virtual optical center O21. The cameras 225 andreflectors REF3 are arranged to divide an entire visual field into twoparts each for reflecting each field to each camera. Each of thesecameras can obtain a half the entire visual field in a square format byone occurrence of exposure. As a result, the composite camera unit PG18can obtain a rectangular omnidirectional view by simply combining thetwo images obtained. Since the camera concentrates optical centers, theprocess has no need to correct an error caused by separation betweenoptical centers.

A photographed image can be mapped on an octahedron to obtain threerectangular images by the method introduced in the third embodiment. Thepyramid's base F18 is not necessarily be a square. For example, apyramid with rectangular base with the aspect ratio of 3:2 or 9:8provides eventually a rectangular image with 3:4 or 9:16 aspect ratio tofit in a wide-use display monitor. A pyramid with rectangular base withthe aspect ratio of 1:2 provides eventually a square image. Thereflector can be replaced by any arbitrary 3-D objects other than asquare pyramid, such as other polyhedron including (semi) regularpolyhedron, open surfaces including hyperboloid, 3-D objects containingcurved surfaces and surfaces of revolution. For example, it is suitablethat geometry of a reflector is designed to correct distortions such assolid angle's distortion while reflecting a view to a camera.

Sixteenth Embodiment

Parallax between optical centers in a composite camera generates stereophotos. FIG. 49 is a top view of a cubic composite camera PG6. Each lens216 along each optic axis covers 2π sr visual field 215. A parallax 181between adjacent two lenses 216 generates a stereoscopic visual field217 captured by the lenses.

The term of “lens” used in this explanation pertains to any opticalelement for converging light such as a pinhole. A plurality of cameras(incl. a couple of lenses for a parallax) can share a common plane ofprojection. For example, a plane can receive exposures by these opticalcenters and/or optic axes one after another. A plane of projectionincludes a 3-D surface composed of a plurality of faces. For example, anexposure on ellipsoid's surface from one of two focuses can avoiddiffused reflections. The note in this paragraph is not only for the16th embodiment but also for the all embodiments.

Seventeenth Embodiment

The invention includes a photographing method that considers an exposureas a mapping process and a plane for placing a film, image sensorincluding any other photosensitive components thereon as a surface formapping. This embodiment illustrates a plurality of plane of projectioncomposing a polyhedral surface. Most of existing cameras have a film orimage sensor on a flat plane. For carrying out the invention with anexisting camera it needs an image processing to modify and adjust anobtained omnidirectional image to a rectangle. The process causes a lossin an image's quality or delay for displaying the image in case ofshowing an animation. The method places image sensors on a preferablepolyhedron and simplifies the process by directly exposing, that ismapping, an image on the polyhedral surface of projection.

FIG. 42 shows a tetrahedral camera PG9 with a lens in this case apinhole 189 on each tetrahedral face. The camera arranges planes ofprojection parallel to the tetrahedral faces. A plane F8 is one of theplanes of projection. Four optic axes are equally distributed and faceto four planes of projection and cover an entire visual field. Thus, thecamera PG9 can obtain a tetrahedral image by a simple exposure withoutcomplex image processing. An exposure especially by a pinhole is a pureoptical projection without a distortion that an ordinary lens tends tocause. A mechanism of a camera with a pinhole is simple, which is ableto minimize a size of composite camera. However, a resolution quality islow by a pin whole camera. By dividing an entire visual field, aplurality of cameras with a pinhole can accumulate resolutions and acomposite camera can maintain a high resolution image. Three innerpyramid surfaces defined by a tetrahedral center O17 and vertices of thetetrahedron PG9 can replace a plane of projection F8 such that eachcamera can photograph almost 2π sr visual field.

Eighteenth Embodiment

The invention includes a method in which an outline of a plane ofprojection fits to an image circle. FIG. 42 shows a composite cameraPG10 with an outline of a truncated tetrahedron that truncates itsvertices and edges. The outline of its face forms a hexagon. Thus, thecamera PG10 mounts a hexagonal plane of projection F11 according to atetrahedron. The hexagonal plane fits to an image circle C3 well.Truncated vertices F10 and edges F9 can mount output (input) connectorsor sockets for power 193, thumbscrews of tripods 194 and/or slots fordata storage devices 195. It is difficult to keep these necessarycomponents away from an entire visual field when upon a single exposure.The camera PG10 hides them by arranging them along edges and vertices ofa polyhedron that defines its camera alignment.

Nineteenth Embodiment

The invention includes a mapping method with a much subdivided iso-areagrid on a plane of projection. The grid determines shapes andarrangements of image sensors such as CCD or CMOS and electro-opticmodulators (EOM) such as LCD or CRT Many existing cameras arrangerectangular image sensors along a rectangular grid. The arrangement doesnot match to a circulation for reading out and does not evenly arrangecolor filters for three primary colors. The method adjusts the imagesensor's shape to a hexagon or other polygons to evenly arrange imagesensors along an iso-area grid.

FIG. 43 closes up an image sensor's arrangement 203. A subdividediso-area grid G13 arrays image sensors 205. Regions 206 between imagesensors provide circulations for reading out. Output efficiency by ahexagonal image sensor is better than by a rectangular sensor since itfits to a circle of confusion. Symbols R, G and B stand for threeprime-color filters that are arranged evenly along a three-way grid by adense iso-area division. It illustrates that a hexagonal image sensorand a three-way grid fit better in a tetrahedral or other polyhedralcomposite camera with a triangular face and minimize a quality loss thana rectangular image sensor and grid.

Another arrangement 204 is obtained by rotating each sensor 205. A roundimage sensor 205 a can replace the sensor 205. A region 206 between thesensors provides a circulation for reading out. Its hexagonal shape andthe three-way-gird can be other polygonal shape and/or other gridpattern. For example, a pentagonal sensor and a pentagonal grid patternfit to a dodecahedral composite camera. The image sensor can be replacedby other sensors for another electromagnetic or a sound wave. A gridother than an iso-area grid can be used for an arrangement of thesensor. The method can be applied for the shape and an arrangement ofelectro-optic modulators (EOM) such as LCD or CRT. It is preferable forboth optical modules to correspond in shape and arrangement.

Twentieth Embodiment

The invention includes a method with a 3-D surface of projection foreach optic axis composed of a plurality of planes. On one hand, it issuitable that an ordinary camera has a plane of projection for rollingup a stripe of film. On the other hand, in such a mechanism, there is adifference in distance from its focal point to a plane of projection atits center and its periphery. That causes a critical over exposure atits center if an image at peripheries needs be exposed properly in awide-angle photo. It needs an adjustment by a complex composition ofseveral lenses to solve it. Instead, the method provides a 3-D surfaceof projection that arrays distances from its focal point to every pointon the surface. This method is explained by taking a cubic compositecamera as an example.

FIG. 41 schematically shows a composite camera PG6 arraying an opticaxis AX4 at the center of each cubic face. A lens, in this case apinhole is positioned on the axis AX4 in particular at the intersection178 with a cubic surface. Thus, the camera divides an entire view intosix parts. A square pyramid of projection F6 is composed of fourtriangles defined by a cubic center O16 and midpoints 172 of segmentsconnecting the center O16 and cubic vertices V16. A hatched region isone of the four triangles.

Four triangles F7 defined by a cubic center O16 and cubic vertices V16can compose a 3-D surface of projection instead. The arrangement with anaccurate pinhole can capture almost a 2π sr view. Each obtained imagesby each axis is wider than an assigned view therefore overlaps toadjacent images. The overlapping visual field generates a stereo photo.A square defined by points 172 can be a part of a 3-D surface ofprojection. A cross section 179 of the camera shows that such a 3-Dsurface of projection reduces differences in distance from its focalpoint to points on the surface.

In FIG. 57, exposed images on 24 planes of projection are assembled in apolyhedron PG7 with 24 faces. An obtained image by a plane F6 is mappedon a square pyramid F6 b defined by its apex 188 and other vertices 184.The obtained images can be assembled in a similar polyhedron, a rhombicdodecahedron too. The polyhedral image is iso-area-mapped on arectangular plane possibly through multi-layer-mapping on a cube,tetrahedron or octahedron. Images obtained by the planes F7 areassembled on an octahedron PG8 and flattened out on a square defined byits vertices 180.

The method can obtain several types of polyhedral images and choose anoptimum rectangular image. Simple explanation for better understandingneeds geometrical term. Therefore, for practicing the 20th embodiment itallows an error by a parallax owing to the volume of camera as long asit obtains the same effect. The note in this paragraph is not only forthe 20th embodiment but also for the all embodiments.

Twenty-first Embodiment

The invention relates to mapping a 3-D surface on a rectangle. Theinvention includes a method to integrate the 3-D arrangement of plane ofprojection on a rectangular plane. The embodiment illustrates anoctahedral composite camera, which simplifies a complex arrangement byintegrating the surfaces on three square planes.

FIG. 44 shows an octahedral composite camera PG11. A lens, which is apinhole 207 in this case, is mounted on an octahedral face. Planes ofprojection are mounted on a part of inner faces of a triangle pyramidPG25 defined by an octahedral face as its base and by the center of theoctahedron as its apex. It shows that eight surfaces of projection areintegrated on three inscribed squares F13 defined by octahedral verticesV25. Thus, image sensors can be easily arranged on the squares, whichprovide eight pyramid's surfaces of projection for eight optical axes.The method can be applied to other 3-D object as long as it obtains thesame effect.

An octahedral camera can obtain an entire view with image sensors on apyramid surface F12 that capture ½π sr view by a lens 207. It can obtaintwo tetrahedral entire views with a larger pyramid F12 a by which eachlens captures 1π sr view. It can obtain four sets of entire views withimage sensors mounted on entire squares by which each lens captures 2πsr view. Otherwise, it can generate a stereo photo by overlapping imagesobtained by the larger surfaces of projection.

Planes F13 a including F13 can be utilized as reflectors. The planescross at right angles and are able to reflect lights from any directionsto where it comes from. Thus, cameras with the reflector can quicklysurvey relational position each other by utilizing the principle. Thismethod is applicable only in case that each surface of projection covers½π sr view. In case of wider view, the reflector disturbs a visualfield. Planes F13 a can be utilized for stabilizing the camera. Theoctahedral camera illustrates a 3-D surface of projection in thisembodiment but can mount a plane of projection parallel to an octahedralface as well.

Twenty-second Embodiment

The invention includes a method to cover a desired entire space bynetworking a plurality of cameras. A single standpoint of a cameraoccasionally has a blind zone if the desired space shapes complicated.Such space needs to distribute a plurality of visual fields to coveroverall space. The method can quickly survey relational position ofcameras.

FIG. 45 schematically shows a cuboctahedral composite camera PG12 withsix optic axes on each cuboctahedral squares F25. Six optic axes equallydivide an entire view into six visual fields. A light source and atriangular pyramid of reflector 208 are mounted on each triangular faceF24. Eight reflectors 208 compose an omnidirectional reflector. Itspyramid surfaces meet at right face angles.

In case of a network by distributing the cameras in an observed space,the reflector can capture light sources coming from other cameras andsurvey each camera's positional relation. Thus, a desired space iscovered by a plurality of omnidirectional views from differentstandpoints while recognizing their positional relation.

The cameras can be arranged to surround a particular subject to observeit from many sides. The method can be specialized in capturing otherportion of the electromagnetic spectrum than visible light, such asradio wave, infrared, ultraviolet, x ray and gamma ray. The method canbe applied to MRI and also to sonar for capturing sound wave. The methodcan display other image than ordinary scenery, such as thermograph,magnetic field and Doppler effect. For the method, a simultaneousexposure is suitable for capturing many moving object. Meanwhile a lessnumber of cameras can capture each assigned visual field by rotating it.In this case, it can concentrate each optical center of each exposure onits rotation axis. Same effect can be obtained by rotating videoequipment while recording. The note in this paragraph is not only forthe twenty-second embodiment but also for the all embodiments.

The method can customize an arrangement of optic axis, reflector, andlight source. Fourteen axes on every faces generate an omnidirectionalcuboctahedral image. The cuboctahedral image can iso-area-map on twosides of an inscribed square 209 or a hexagon 210 defined bycuboctahedral vertices V26. Both square and hexagonal images can bearranged on a plane with some gaps with seamless continuations orwithout any gaps with discontinuous seam.

Twenty-Third Embodiment

The invention includes a method to vary arrangements of optic axes andto generate various omnidirectional images for different needs such asneeds for a stereo effect. FIG. 47 shows a composite camera unit PG14composed of two cubic camera modules M1 with a lens 219 or a pinhole.The modules M1 are coupled together by a hinge 249, and can rotate aboutan axis AX6. The hinge is preferably configured to fix the modules M1 atevery rotation of certain angles. A shutter 220, a battery unit and/or asocket for power 221, a display unit 218 and connector 248 are mountedon other cubic faces. A connector 248 connects to other coupling cameraPG14. A connector 248 can preferably function as an output (input)connector and/or a thumbscrew of a tripod. In this illustration, aspherical surface of projection 222 is mounted on the camera module M1.In case with a pinhole at the center of the sphere, the surface keeps adistance to a focal point equally. An incoming light is projected on thesurface at right angle. These features are suitable for a properexposure. The 3-D surface of projection tends to cause the diffusedreflection problem. The spherical surface reflects the light back to thepinhole, which keeps off a diffused reflection inside the device.

FIG. 48 shows a mode of use, 253, of the composite camera unit PG14 inwhich the two camera modules M1 are coupled to align their optical axesin a common direction for capturing a stereoscopic photo and a mode ofuse, 254, of the same composite camera unit PG14 in which the two cameramodules M1 are coupled to orient their optical axes in oppositedirections for capturing an omnidirectional view.

FIG. 50 shows an arrangement with three cameras PG14. In each cameraPG14, two axes of devices M1 cross at right angle. Three cameras PG14are integrated to capture an entire view in six divisional regions bysetting six optical axes in accordance with a cube PG15. Thisarrangement assigns a device M1 one of six divisional views. Thisarrangement captures an entire view with higher resolution than one by acamera. Cube's niches at bottom and on top can preferably mount a moduleM2, such as a pan head, a light source or other essential modules.Truncated vertices 224 at the bottom stabilize the camera. A parallaxbetween optical centers generates a stereo photo if lens 219 covers awider visual field than an assigned visual field. User can customize anarrangement of the camera modules M1 for a different resolution and fora different setup of parallax. User can customize an arrangement ofcomponents on device's faces such as mounting more connection 248 orreplacing it to a hinge 249. A camera PG14 can be composed of othercombination of device M1.

Twenty-Fourth Embodiment

The invention includes a method to vary arrangements of optic axes toprovide choices of polyhedral images. The invention also includes amethod to mount storage, circuit and other necessary components whilemodifying a shape of the composite camera without disturbing itsgeometrical attribute.

FIG. 46 schematically shows a rhombicuboctahedral composite camera PG13and its section 14 a. The rhombicuboctahedron is obtained by truncatingedges and vertices of an octahedron 211. In one of arrangements, thecamera PG13 integrates image sensors for all optic axes on six octagonsF14 defined by six vertices V27. The axes are assigned according toeight triangular faces. For example, a hatched region F15 is one ofcomposed planes of projection. The arrangement can obtain an entireoctahedral image that is derived from the arrangement of eight opticaxes. A RAM, storage, power source and other necessary components areinstalled in a cubic space 212 created by the six octagons F14.Removable additional components such as component cameras are installedin niches 213.

The camera can arrange optic axes to obtain diverse polyhedral images.An arrangement of 26 optic axes towards its all faces generates arhombicuboctahedral image. A lens with an ordinary angle θ5 can capturean assigned divisional visual field according to the arrangement. Anarrangement of optic axes towards twelve squares F17 obtains a rhombicdodecahedral image. The rectangles F17 are sections by truncating edgesof an octahedron 211. An arrangement of optic axes towards sixrectangles F16 obtains a cubic image. The rectangles F16 are sections bytruncating vertices of an octahedron 211. This arrangement can mount thereflectors on regions F15 to compose an omnidirectional reflector andlight sources, tripod, an output (input) connector, an antenna on otherfaces. The rhombicuboctahedral camera contains three sets of eight sidedprisms. Each prism's base crosses with an axis AX6 at right angle. Anarrangement of optic axes on its eight side faces generates up to threeprismatic and/or cylindrical images, which can generate up to threepanorama images, an accustomed representation.

Twenty-Fifth Embodiment

The invention includes a composite mapping method that combinesdifferent types of mapping such as a mapping with a gnomonic projectionalong an ordinate and orthographic projection along an abscissa. Themethod is available for 3-D objects such as with open or curved surfacesor 3-D objects by uniting several objects. It illustrates a mappingmethod that combines cylinders and extracts distorted parts from thecylinders that compose a 3-D object with closed surface.

A schematic diagram shown in FIG. 53 illustrate a process for obtaininga square image from an object generated by intersecting the twocylinders PG19. An entire view is mapped on a cylinder PG19 by agnomonic projection along longitude and an orthographic projection alonglatitude. A hatched region F19 is an overlapped region by intersectingtwo such cylinders. An entire view on a 3-D object PG20 described with aset of graticule is obtained by four such regions F19 and by removingfour other regions PG20 a. A square F20 receives mapping from a 3-Dobject on its both sides on top and bottom along axis AX8. A rectangularimage is obtained by integrating the images on its two sides.

Cylindrical projection is known as a panorama photography and Mercatorprojection. The method is composed of such mappings with cylinders.Therefore, its appearance is familiar for user who has been familiarwith a panorama and Mercator projection. Meanwhile it reducesdistortions on a rectangular image by extracting less distorted part incylindrical projection. The method can be applied to other 3-D objectsdefined by curved surfaces, straight lines and curves. An objectcomposed of corns can obtain an image associated with a conicprojection.

Twenty-Sixth Embodiment

The invention includes another mapping method compromising an existingcylindrical projection and the rectangular image according to theinvention. FIG. 54 shows a rhombicuboctahedron PG23 created by anintersection of another cylinder and the object PG20 in FIG. 53.Overlapped parts of cylindrical surface compose congruent twelverhombicuboctahedral surfaces defined by arcs 238, connecting verticesV29. Straight lines subdivide the object into 24 regions. The polyhedronwith the 24 curved surfaces fits to a cube, octahedron and tetrahedronby sharing many vertices, faces and edges. Therefore, it is suitablymapped on these polyhedrons. For example, a hatched region 240 is mappedon an octahedral face.

Specifically, A simple gnomonic projection with these polyhedronsmaintains area ratios. The polyhedron can generate a rectangular imageaccording to the invention. Thus, the method intermediates between awide used cylindrical projection and the rectangular image by theinvention. Simple explanation for better understanding needs geometricalterm. Therefore, for practicing, the 26th embodiment allows an error inthe polyhedrons and polygons for the mapping as long as it obtains thesame effect. The note in this paragraph is not only for the 26thembodiment but also for other related embodiments.

Twenty-Seventh Embodiment

The invention includes a method with mapping on an object composed ofarcs in a longitudinal direction, which contains one or more rectangularsections. The arcs buffer a mapping from an angle distortion aroundpolyhedral edges since any section of a sphere is circle. The imagerepresents an omnidirectional view without disturbances. FIG. 58schematically shows a 3-D object PG21 divided by a graticule-iso-areagrid G15. Longitudes 226 and latitudes 231 compose the grid G15. A partof the grid in one eighth of an entire visual field is shown in thisdiagram. Arcs 226 radiates from a pole 227 to points on a square's F21edge 228. Curved lines 231 link points that divide arcs 226.

Meanwhile an iso-area grid G16 is on a square F21 a composed of a linescorresponding to its longitudes 226 a, latitudes 231 a and square'sedges 228 a representing an equator. A part of the grid in one eighth ofan entire visual field is shown in this diagram. Segments 226 a radiatesfrom a point representing a pole 227 a to points on the square's edge228. Curved lines 231 a link points that divide the segments 226 a. Eachdivisional region 233 on a grid G15 is iso-area-mapped on eachcorresponding region 233 a on the square. Thus, a square image isobtained. The arcs of 3-D object PG21 connect its surfaces withoutvisible edges. That eliminates an angle distortion around polyhedraledges when displaying on a square.

Twenty-Eighth Embodiment

The invention includes an iso-area dividing method by curves such asconic sections. FIG. 59 schematically shows a grid G17 in a squarecovering an entire visual field. It is a modification of a grid shown inFIG. 58. The grid G16 shown in FIG. 58 corresponds to a square regionF21 b in this diagram. Segments 226 b, 231 b and a square's edges 228 bcompose a grid G17. Curved segments 226 b such as hyperbolas radiatesfrom a point representing a pole 227 b to points 230 b on the square'sedge 228 b. Curved segments 231 b such as parabolas link points thatdivide the segments 226 b. The method can obtain a conformal mapping byarranging a segment 231 b and 226 b to cross at right angle. Eachdivisional region 233 on a grid G15 is iso-area-mapped on eachcorresponding region 233 b on the square. Continuous curved linesreplace segments bent on the edge 228 b. That eliminates an angledistortion around the edges.

A number of frequencies of the iso-area division can be increased forits accuracy. Meanwhile it can be decreased down to a simple tetrahedronor octahedron for a rapid image processing though a correction ofdistortion is limited. It is preferable to divide segments and/or angleθ6 at their intersections as equally as possible so that obtained imageis nearly an equidistant and/or conformal in case of a world map. Thegrid G16 and G17 can be obtained by an optical projection of the gridG15 from a point on the axis AX9. The curved lines of the grid includesgeodesic lines, arcs, Bézier curves, spline curves, parabolas, otherconic curves, elliptic curves, other cubic curves, Cassinian ovals,Clothoid curves and lines made by combining these curves. The note inthis paragraph is not only for the 28th embodiment but also for the allembodiments.

Twenty-Ninth Embodiment

The invention includes a method of iso-area division using anintermediate object between a sphere, a polyhedron and a rectangularplane. FIG. 55 schematically shows a process of multi-layer-mapping ofan octahedral image on a square while keeping the displayed image fromdisturbances. The method maps the octahedral image PG22 on an objectPG22 a composed of curved surfaces and then on a square F22. Itillustrates a half part of an entire visual field such as four of eightoctahedral faces. A grid G18 divides the shown four faces into 24regions. A region 235 is one of them. The grid G18 is composed offollowing two elements: (1) segments 236 connecting octahedral verticesV28 and their midpoint 234 and (2) its edges 237.

The object PG22 a shares the center O22 and all vertices V28 except avertex V28 a with the octahedron PG22. The grid G18 a equally dividesthe shown four faces into 24 regions. A region 235 a is one of them. Thegrid G18 a is composed of following three elements: (1) segments 237 aradiating from the vertex V28 a to other vertices V28, (2) segmentsradiating from the vertex V28 a to midpoints 234 of straight segmentconnecting vertices V28 (3) segments connecting two vertices V28 passingthrough a point on a segment 237 a. A segment 236 a is a curved linesuch as linked arcs. Its curvature is adjusted so that any region is notmapped outside the square frame later. The square F22 shares thevertices V28 and the center O22 with the octahedron PG22. A grid G18 bequally divides the square into 24 regions. A region 235 b is one ofthem. The grid G18 b is composed of following three elements: (1)segments 237 b radiating from the center O22 to vertices V28, (2)segments radiating from the center O22 to midpoint 234 of the edges and(3) segments connecting two vertices V28 passing through a point on asegment 236 b.

Thus, an octahedral image PG22 is iso-area-mapped on a square F22 via anobject PG22 a. The method can omit the mapping process on an object PG22a as long as it obtains the same effect.

Thirtieth Embodiment

The invention includes a method of downsizing a camera used to capturean entire view in small and/or narrow space. FIG. 56 schematically showsa cylindrical composite camera unit PG24. In this camera unit PG24, twocameras 241 are arranged in a face-to-face relation along an axis AX11of the cylinder. Each camera 241 comprises a lens 242 and a conicreflector 244 having the lens 242 on its top. The reflector 244 reflectsa nearly hemispherical view (horizontally 360° and vertically θ8) to thelens 242. Therefore, the lens's angle θ6 of view does not need to bewide. If the angle is wider as much as θ7, the surplus angle of view isadded to the other lens's 242 angle of visual field. The angle θ9 of theconic reflector and a distance L5 between cameras define its angle ofview. Therefore, the transparent cylinder 245 slides and adjusts thedistance L5 for proper angle, for zooming and for minimizing camera'ssize PG24 a.

Thus, an omnidirectional image taken by the two cameras can berectangulated by the mapping method according to the invention.Meanwhile it can carefully observe a desired subject such as a tumorwith a stereoscopy by utilizing a parallax between two optical centers246. The reflector 244 of each camera 241 has a configuration capable ofaccommodating the counterpart camera 241 and thereby contributes todownsizing the composite camera unit PG24. Further, this configurationof each reflector 244 contributes to provide a distance necessary forfocalization when the camera is used in a narrow space.

A cone-shaped reflector 244 is preferable for converting an obtainedimage to a panorama in case of observing inside of a tube such asintestine. A hatched space 247 can store a data storage device, output(input) connector and/or an additional camera 241 a. The camera 241 acovers a dead angle θ60 hided by the camera itself. In this case, fourcameras 241 are incorporated in the composite camera unit PG24 withtheir optical axes aligned in parallel to the axis AX11 and do notrequire an increase of the diameter D1 of the cylinder to provide aminimum focal distance.

1. An information processing method for transferring information held ona start face having a plurality of start faces that are contiguous andeach defined by one or more lines and points onto an end face having aplurality of end faces that are contiguous while maintaining one-to-onecorrespondence between the information on said start face and said endface, comprising: at least one of said start face and said end facebeing a rectangular plane; each said divisional start face beingdeformed to just fit in each said divisional end face without gaps; eachof said lines defining said divisional start faces being maintained aslines defiling a corresponding one of said divisional end faces; each ofsaid points defining said divisional start faces being maintained aspoints defining correspondent one of said divisional end faces;adjacency of every two adjacent ones of said divisional start facesbeing maintained as adjacency of every two corresponding ones of saiddivisional end faces; and a first area ratio of each of said divisionalstart faces relative to the entirety of the start face is substantiallyequal to a second area ratio of each of said divisional end facesrelative to the entirety of the end face.
 2. The information processingmethod according to claim 1, wherein the entirety of said divisionalstart faces forms a 3-D surface.
 3. The information processing methodaccording to claim 2, further comprising: an additional process forsubdividing at least a part of said divisional start faces into aplurality of contiguous subdivisional faces or integrating at least apart of said divisional start faces to one or more contiguous largerdivisional faces.
 4. The information processing method according toclaim 1, further comprising an additional process for arranging aplurality of said rectangular planes in at least one of a horizontaldirection and a vertical direction to make a rectangular display planelarger than said rectangular plane and containing no substantialdiscontinuity in information corresponding to said start informationbetween every adjacent rectangular planes.
 5. The information processingmethod according to claim 1, wherein all of said divisional start faceshave a substantially equal area and all of said divisional end faceshave a substantially equal area.
 6. The information processing methodaccording to claim 2, wherein at least two or more adjacent ones of saiddivisional start faces make a substantially equal face angle.
 7. Theinformation processing method according to claim 1, wherein at least apart of interior angles between adjacent ones of the lines defining thedivisional start faces at the point defining the divisional start facesmakes a substantially equal angle, and at least a corresponding part ofinterior angles between adjacent ones of lines defining the divisionalend faces at the point defining the divisional end faces makes asubstantially equal angle.
 8. The information processing methodaccording to claim 1, wherein at least a part of the lines defining thedivisional start faces has a substantially equal length and at least apart of the lines defining the divisional end faces has a substantiallyequal length.
 9. The information processing method according to claim 2,wherein at least some of the lines defining the divisional start facesare linked together to form loops which are substantially parallel toeach other, and at least some of lines defining said divisional endfaces are linked together to form loops that are substantially parallelto each other.
 10. The information processing method according to claim1, wherein at least a part of said lines defining the divisional startfaces is curved lines, and/or at least a part of lines defining thedivisional end faces is curved lines.
 11. The information processingmethod according to claim 1, wherein said start face is an entiresurface of a polyhedron, and at least some of the lines defining thedivisional start faces are edges of the polyhedron.
 12. The informationprocessing method according to claim 2, wherein at least a part of thestart faces defines a curved surface.
 13. The information processingmethod according to claim 4, wherein a plurality of the rectangularplanes are tessellated vertically and/or horizontally with no gaps. 14.The information processing method according to claim 4, wherein thepolyhedron is a tetrahedron, and each said divisional start face definedby lines including tetrahedral edges and points including tetrahedralvertices is deformed to just fit each divisional end face defined bylines and points including lines corresponding to the tetrahedral edgesand points corresponding to the tetrahedral vertices on a rectangulardevelopment of the tetrahedron, which forms said rectangular plane. 15.The information processing method according to claim 3, furtherincluding an additional process for integrating the divisional startfaces onto an intermediate polygonal plane defined by two separatepoints on one of said divisional start faces and at least one point onanother of said divisional start faces.
 16. The information processingmethod according to claim 3, further including an additional process forintegrating the divisional start faces onto an intermediate polygonalplane defined by midpoints of at least three of the lines definingdivisional start faces.
 17. The information processing method accordingto claim 2, wherein said divisional start faces comprises planes ofprojections of a plurality of cameras arranged to constitute thedivisional start faces.
 18. The information processing method accordingto claim 1, wherein a plurality of image sensors and/or electro-opticmodulators are arranged to constitute the divisional start faces and/orthe divisional end faces.
 19. The information processing methodaccording to claim 4, further including an additional process fordisplaying and outputting a cut of the information mapped on saidrectangular plane by outlining it by a polygonal viewer that can berotated, slid and shifted to select any desired cut of the information.